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The Geometry of Characters of Hopf Algebras 3 The geometry of characters of Hopf algebras Geir Bogfjellmo and Alexander Schmeding Abstract Character groups of Hopf algebras can be used to conveniently describe several species of “series expansions” such as ordinary Taylor series, B-series, aris- ing in the study of ordinary differential equations, Fliess series, arising from con- trol theory and rough paths. These ideas are a fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups. Our main result is the construction and study of Lie group struc- tures for Hopf algebras which are graded but not necessarily connected (in the sense that the degree zero subalgebra is finite-dimensional). Key words: infinite-dimensional Lie group, pro-Lie group, character group of a Hopf algebra, regularity of Lie groups, continuous inverse algebra MSC2010: 22E65 (primary); 16T05, 43A40, 58B25 (Secondary) 1 Foundations: Character groups and graded algebra . ...................... 4 2 Geometry of groups of characters . ..................... 7 3 Appendix: Infinite-dimensional calculus . ....................... 18 4 Appendix: Characters and the exponential map . ..................... 21 References ......................................... ............................ 23 arXiv:1704.01099v4 [math.GR] 27 Sep 2017 Geir Bogfjellmo Chalmers tekniska h¨ogskola och G¨oteborgs universitet, Matematiska vetenskaper, 412 96 G¨oteborg, Sweden e-mail: [email protected] Alexander Schmeding NTNU Trondheim, Institutt for matematiske fag, 7491 Trondheim, Norway e-mail: [email protected] 1 2 Geir Bogfjellmo and Alexander Schmeding Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory [14], numerical analysis [10] and the theory of regularity structures for stochastic partial differential equations [25]. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coal- gebra that is also equipped with an antiautomorphism known as the antipode, sat- isfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra mor- phisms from a Hopf algebra H to a commutative algebra B. Examples include or- dinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. An important fact about such algebraic objects is that, if B is commutative, the set of algebra mor- phisms Alg(H ,B), also called characters, forms a group with product given by convolution a ∗ b = mB ◦ (a ⊗ b) ◦ ∆H . These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combina- torics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups. Topological and manifold structures on these groups are important to applica- tions in the various fields outlined above. In many places in the literature the char- acter group is viewed as “an infinite dimensional Lie group” and one is interested in solving differential equations on these infinite-dimensional spaces (we refer to [6] for a more detailed discussion and further references). This is dueto the fact that the character group admits an associated Lie algebra, the Lie algebra of infinitesimal characters1 g(H ,B) := {φ ∈ HomK(H ,B) | φ(xy)= φ(x)εH (y)+ εH (x)φ(y), ∀x,y ∈ H }, whose Lie bracket is given by the commutator bracket with respect to convolution. As was shown in [5], character groups of a large class of Hopf algebras are infinite- dimensional Lie groups. Note however, that in ibid. it was also shown that not every character group can be endowed with an infinite-dimensional Lie group structure. In this note we extend these results to a larger class Hopf algebras. To this end, recall that a topological algebra is a continuous inverse algebra (or CIA for short) if the set of invertible elements is open and inversion is continuous on this set (e.g. a Banach algebra). Then we prove the following theorem. 1 Note that this Lie algebra is precisely the one appearing in the famous Milnor-Moore theorem in Hopf algebra theory [41]. The geometry of characters of Hopf algebras 3 H H H Theorem A Let = ⊕n∈N0 n be a graded Hopf algebra such that dim 0 < ∞ and B be a commutative CIA. Then G (H ,B) is an infinite-dimensional Lie group whose Lie algebra is g(H ,B). As already mentioned, in applications one is interested in solving differential equations on character groups (see e.g. [42] and compare [6]). These differen- tial equations turn out to be a special class of equations appearing in infinite- dimensional Lie theory in the guise of regularity for Lie groups. To understand this and our results, we recall this concept now for the readers convenience. Regularity (in the sense of Milnor) Let G be a Lie group modelled on a locally convex space, with identity element e, and r ∈ N0 ∪{∞}. We use the tangent map of the left translation λg : G → G, x 7→ gx by g ∈ G to define g.v := Teλg(v) ∈ TgG for r r v ∈ Te(G)=: L(G). Following [20], G is called C -semiregular if for each C -curve γ : [0,1] → L(G) the initial value problem η′(t) = η(t).γ(t) (η(0) = e has a (necessarily unique) Cr+1-solution Evol(γ) := η : [0,1] → G. If furthermore the map evol: Cr([0,1],L(G)) → G, γ 7→ Evol(γ)(1) is smooth, G is called Cr-regular.2 If G is Cr-regular and r ≤ s, then G is also Cs- regular. A C∞-regular Lie group G is called regular (in the sense of Milnor)–a property first defined in [40]. Every finite-dimensional Lie group is C0-regular (cf. [43]). In the context of this paper our results on regularity for character groups of Hopf algebras subsume the following theorem. H H H Theorem B Let = ⊕n∈N0 n be a graded Hopf algebra such that dim 0 < ∞ and B be a sequentially complete commutative CIA. Then G (H ,B) is C0-regular. Recently, also an even stronger notion regularity called measurable regularity has been considered [19]. For a Lie group this stronger condition induces many Lie theoretic properties (e.g. validity of the Trotter product formula). In this setting, L1- regularity means that one can solve the above differential equations for absolutely continuous functions (whose derivatives are L1-functions). A detailed discussion of these properties can be found in [19]. However, we will sketch in Remark 19 a proof for the following proposition. H H H Proposition C Let = ⊕n∈N0 n be a graded Hopf algebra with dim 0 < ∞ which is of countable dimension, e.g. H is of finite type. Then for any commutative Banach algebra B, the group G (H ,B) is L1-regular. One example of a Hopf algebra whose group of characters represent a series ex- pansion is the Connes–Kreimer Hopf algebra or Hopf algebra of rooted trees HCK. Brouder [10] established a very concrete link between HCK and B-series. B- series, due to Butcher [12], constitute an algebraic structure for the study of inte- 2 Here we consider Cr ([0,1],L(G)) as a locally convex vector space with the pointwise operations and the topology of uniform convergence of the function and its derivatives on compact sets. 4 Geir Bogfjellmo and Alexander Schmeding grators for ordinary differential equations. In this context, the group of characters G (HCK,R) is known as the Butcher group. The original idea was to isolate the numerical integrator from the concrete differential equation, and even from the sur- rounding space (assuming only that it is affine), thus enabling a study of the integra- tor an sich. Another example connecting character groups to series expansions arises in the theory of regularity structures for stochastic partial differential equations (SPDEs) [25, 11]. In this theory one studies singular SPDEs, such as the continuous parabolic Anderson model (PAM, cf. the monograph [33]) formally given by ∂ − ∆ u(t,x)= u(t,x)ζ(x) (t,x) ∈]0,∞[×R2, ζ spatial white noise. ∂t We remark that due to the distributional nature of the noise, the product and thus the equation is ill-posed in the classical sense (see [25, p.5]). To make sense of the equation, one wants to describe a potentialsolution by “local Taylor expansion” with respect to reference objects built from the noise terms. The analysis of this “Taylor expansion” is quite involved, since products of noise terms are not defined. However, it is possible to obtains Hopf algebras which describe the combinatorics involved. Their R-valued character group G is then part of a so called regularity structure (A ,T ,G ) ([11, Definition 5.1]) used in the renormalisation of the singular SPDE. See Example 25 for a discussion of the Lie group structure for these groups. 1 Foundations: Character groups and graded algebra In this section we recall basic concepts and explain the notation used throughout the article. Whenever in the following we talk about algebras (or coalgebras or bial- gebras) we will assume that the algebra (coalgebra, bialgebra) is unital (counital or unital and counital in the bialgebra case).
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